What is the equation to update the weights in the perceptron algorithm?
I will tell you my knowledge, correct me if I am wrong.
Perceptron Learning Algorithm (PLA) is a simple method to solve the binary classification problem.
Define a function:
$$
f_w(x) = w^Tx + b
$$
where $x \in \mathbb{R}^n$ is an input vector that contains data points and $w$ is a vector with the same dimension as $x$ which present for the parameters of our model.
Call $y=label(x)=\{1,-1\}$ where $1$ and $-1$ are the label of each $x$ vector.
The PLA will predict a class like this:
$$
y=label(x)=sgn(f_w(x))=sgn(w^Tx+b)
$$
(The definition of sgn function can be found in this wiki)
We can understand that PLA tries to define a line (in 2D, or a plane in 3D, and hyperplane in more than 3 dimensions coordinate, I will assume it in 2D from now on) which separate our data into two areas. So how can we find that line? Just like every other machine learning problems, define a cost function, then optimize the parameters to have the smallest cost value.
Now, let define the cost function first, you can see that if a data point lies in the correct area, $y$ and $f(x)$ have the same sign, which means $y(w^Tx+b) &gt; 0$ and otherwise. Similar to your example, I will define:
$$
g(x)=y(w^Tx+b)
$$
We ignore all the points in the safe zone ($g(x)&gt;0$), only update to rotate or move the line to adapt with the misclassified points ($g(x)\le 0$), here, you can understand why we only update if $g(x)\le0$.
We need to define a cost function to minimize it, so our cost function will become:
$$
L(w)=\displaystyle\sum_{x_i\in U}(-y_i(w^Tx_i+b))
$$
where

$U$is the set of the misclassified points
$y_i$ is the label of data point $i$-th
$x_i$ is the $i$-th data vector
$w$ and $b$ is parameters of our model

For each data point, we have the derivative is
$$
\frac{\partial L}{\partial w} = -y_ix_i \\
\frac{\partial L}{\partial b} = -y_i
$$
Finally, update them by Stochastic gradient descent (SGD), we get:
$$
w = w - \frac{\partial L}{\partial w} = w + y_ix_i \\
b = b - \frac{\partial L}{\partial b} = b + y_i
$$
For your last question, notice that the weight and bias changed from $4$-th updated, so we have:
$$
y_5 = 1, x_5 = (4,8), w = (-2,-2), b = -2 \\
\Rightarrow g_5 = +1 \times (4\times(-2) + 8\times(-2)+ (-2)) = -8 -16 -2
$$